Skeletons of near-critical Bienaym\'{e}-Galton-Watson branching processes

TL;DR

This paper studies the structure of near-critical Bienaymé-Galton-Watson branching processes by defining skeletons through particle marking, revealing their asymptotic behavior as birth-death processes and analyzing extinction escape times.

Contribution

It introduces a new skeleton definition for near-critical processes using marking, extending previous models to include subcritical, critical, and supercritical cases with asymptotic analysis.

Findings

Skeletons approximate birth-death processes under rare marking

Limit skeletons for mutation models are derived

Density functions for escape times from extinction are computed

Abstract

Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever. In the critical case it was earlier suggested to distinguish the particles with the total number of descendants exceeding a certain threshold. These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaym\'{e}-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes which can be subcritical, critical or supercritical. We obtain the limit skeleton for a sequential mutation model and compute the density distribution function for the time to escape from extinction.